text: For incompressible flow, the continuity equation and Navier–Stokes equations are filtered, yielding the filtered incompressible continuity equation,
  and the filtered Navier–Stokes equations,
  where  is the filtered pressure field and  is the rate-of-strain tensor evaluated using the filtered velocity.  The nonlinear filtered advection term  is the chief cause of difficulty in LES modeling.  It requires knowledge of the unfiltered velocity field, which is unknown, so it must be modeled.  The analysis that follows illustrates the difficulty caused by the nonlinearity, namely, that it causes interaction between large and small scales, preventing separation of scales.
  The filtered advection term can be split up, following Leonard (1975), as:
  where  is the residual stress tensor, so that the filtered Navier-Stokes equations become
  with the residual stress tensor  grouping all unclosed terms.  Leonard decomposed this stress tensor as  and provided physical interpretations for each term.  , the Leonard tensor, represents interactions among large scales, , the Reynolds stress-like term, represents interactions among the sub-filter scales (SFS), and 
  where  is the diffusive flux of , and  is the sub-filter flux for the scalar .  The filtered diffusive flux  is unclosed, unless a particular form is assumed for it, such as a gradient diffusion model .   is defined analogously to ,
  and can similarly be split up into contributions from interactions between various scales.  This sub-filter flux also requires a sub-filter model.